Coordinate transformations

From Wikiversity
Jump to navigation Jump to search

Vector Transformation in Two Dimensions[edit | edit source]

In three dimensions, the vector transformation rule is written as

where .

In two dimensions, this transformation rule is the familiar

In matrix form,

Since we are using sines, the direction of measurement of is required. In this case, it is measured counterclockwise.

Tensor Transformation in Two Dimensions[edit | edit source]

In three dimensions, the second-order tensor transformation rule is written as

where .

The Cauchy stress is a symmetric second-order tensor. In two dimensions, the transformation rule for stress is then written as

In matrix form,

Since we are using sines, the direction of measurement of is required. In this case, it is measured counterclockwise.

Tensor Transformation in two Dimensions, the intrinsic approach[edit | edit source]

Let construct an orthonormal basis of the second order tensor projected in the first order tensor

The stress and strain tensors are now defined by :

and

Then once constructs the bound matrix in the orthonormal base

with

the rotation matrix in base.

Example[edit | edit source]

is the rotation along the axis in the : base

The associated rotation in the base is :

The rotation of a second order tensor is now defined by :

Four order tensor[edit | edit source]

The élasticity tensor in the : is defined in the : by

and is rotated by:

Related Content[edit | edit source]

Introduction to Elasticity